Engineering Papers⌕ Search

SEARCH · Engineering Papers

Results for “Equation of State”

Search indexed NASA NTRS and DOE OSTI research on propulsion, heat transfer, battery materials and energy systems. Follow report and document links to the original sources.

Quote a phrase for an exact phrase match. Source license links do not imply unrestricted reuse.

At least 127 records · Page 7
All sourcesNASA NTRSDOE OSTI
NASA NTRS1978

Equation of state fits to the lower mantle and outer core

The lower mantle and outer core are subjected to tests for homogeneity and adiabaticity. An earth model is used which is based on the inversion of body waves and Q-corrected normal-mode data. Homogeneous regions are found at radii between 5125 and 4825 km, 4600 and 3850 km, and 3200 and 2200 km. The lower mantle and outer core are inhomogeneous on the whole and are only homogeneous in the above local regions. Finite-strain and atomistic equations of state are fit to the homogeneous regions. The apparent convergence of the finite-strain relations is examined to judge their applicability to a given region. In some cases the observed pressure derivatives of the elastic moduli are used as additional constraints. The effect of minor deviations from adiabaticity on the extrapolations is also considered. An ensemble of zero-pressure values of the density and seismic velocities are found for these regions. The range of extrapolated values from these several approaches provides a measure of uncertainties involved.

Butler, R.↗
NASA NTRS2014

Elastic Constants of Solids and Fluids with Initial Pressure via a Unified Approach Based on Equations-of-State

The second and third-order Brugger elastic constants are obtained for liquids and ideal gases having an initial hydrostatic pressure p(sub 1). For liquids the second-order elastic constants are C(sub 11) = A + p(sub 1), C(sub 12) = A -- p(sub 1), and the third-order constants are C(sub 111) = --(B + 5A + 3p(sub 1)), C(sub 112) = --(B + A -- p(sub 1)), and C(sub 123) = A -- B -- p1, where A and B are the Beyer expansion coefficients in the liquid equation of state. For ideal gases the second order constants are C(sub 11) = p(sub 1)gamma + p9sub 1), C(sub 12) = p(sub 1)gamma -- p(sub 1), and the third-order constants are C(sub 111) = p(sub 1)(gamma(2) + 4gamma + 3), C(sub 112) = --p(sub 1)(gamma(2) -- 1), and C(sub 123) = --p(sub 1) (gamma(2) -- 2gamma + 1), where gamma is the ratio of specific heats. The inequality of C(sub 11) and C(sub 12) results in a nonzero shear constant C(sub 44) = (1/2)(C(sub 11) C(sub 12)) = p(sub 1) for both liquids and gases. For water at standard temperature and pressure the ratio of terms p1/A contributing to the second-order constants is approximately 4.3 x 10(-5). For atmospheric gases the ratio of corresponding terms is approximately 0.7. Analytical expressions that include initial stresses are derived for the material 'nonlinearity parameters' associated with harmonic generation and acoustoelasticity for fluids and solids of arbitrary crystal symmetry. The expressions are used to validate the relationships for the elastic constants of fluids.

Cantrell, John H.↗
NASA NTRS1990

The equation of state for stellar envelopes. IV - Thermodynamic quantities and selected ionization fractions for six elemental mixes

The free-energy minimization technique in the form developed in the preceding papers in this series is employed to evaluate thermodynamic quantities and ionization fractions on a fine temperature and density grid for six astrophysical mixtures of 15 elements. The mixtures range from that appropriate to super-metal-rich stars, through solar abundance, to that for extreme Population II objects. In this paper, the results for solar abundances are summarized in a form that is illustrative and which facilitates comparison with the results from other equation of state calculations.

Mihalas, Dimitri↗
NASA NTRS1993

High-Pressure Equation of State for Partially Ionic Solids

Recently, we showed that the cohesive energy of partially ionic solids may be characterized by a two-term energy relationship consisting of a Coulomb term arising from the valence-charge transfer delta Z between the atoms, and a scaled universal energy function E(sup *)(a(sup *)), which accounts for the partially covalent character of the bond and for the repulsion between the atomic cores for small R; a(sup *) is a scaled length. Normalized cohesive-energy curves of alkali halide crystals and of Ti and Ag halide crystals were obtained, and the cohesive-energy-curve parameters were used to generate theoretical equation-of-state (EOS) curves for the Li, Na, K, Cs, and Ag halides. Good agreement was obtained with the experimental isothermal compression curves over a wide pressure range (0-90 kbar). In this paper we verify that the cohesive-energy relationship is valid for divalent partially ionic solids; physically reasonable charge-transfer values (1.80 less than delta Z less than 2.0) are obtained for MgO, CaO, and CaS. Next, EOS curves for LiF, NaF, Nal, CsCl, Csl, MgO, CaO, and CaS are generated in terms of the cohesive-energy parameters. These EOS's yield excellent fits to experimental isothermal-compression data and to shock-wave data to very high pressures (P(sub max)= 250-1350 kbar).

Schlosser, Herbert↗
NASA NTRS2004

Future Probes of the Neutron Star Equation of State Using X-ray Bursts

Observations with NASA s Rossi X-ray Timing Explorer (RXTE) have resulted in the discovery of fast (200 - 600 Hz), coherent X-ray intensity oscillations (hereafter, %urstoscillations ) during thermonuclear X-ray bursts from 12 low mass X-ray binaries (LMXBs). Although many of their detailed properties remain to be fully understood, it is now beyond doubt that these oscillations result from spin modulation of the thermonuclear burst flux from the neutron star surface. Among the new timing phenomena revealed by RXTE the burst oscillations are perhaps the best understood, in the sense that many of their properties can be explained in the framework of this relatively simple model. Because of this, detailed modelling of burst oscillations can be an extremely powerful probe of neutron star structure, and thus the equation of state (EOS) of supra-nuclear density matter. Both the compactness parameter beta = GM/c(sup 2)R, and the surface velocity, nu(sub rot) = Omega(sub spin)R, are encoded in the energy-dependent amplitude and shape of the modulation pulses. The new discoveries have spurred much new theoretical work on thermonuclear burning and propagation on neutron stars, so that in the near future it is not unreasonable to think that detailed physical models of the time dependent flux from burning neutron stars will be available for comparison with the observed pulse profiles from a future, large collecting area X-ray timing observatory. In addition, recent high resolution burst spectroscopy with XMM/Newton suggests the presence of redshifted absorption lines from the neutron star surface during bursts. This leads to the possibility of using large area, high spectral resolution measurements of X-ray bursts as a precise probe of neutron star structure. In this work I will explore the precision with which constraints on neutron star structure, and hence the dense matter EOS, can be made with the implementation of such programs.

Strohmayer, Tod E.↗
NASA NTRS1976

The Investigation of Optimal Discrete Approximations for Real Time Flight Simulations

The results are presented of an investigation of discrete approximations for real time flight simulation. Major topics discussed include: (1) consideration of the particular problem of approximation of continuous autopilots by digital autopilots; (2) use of Bode plots and synthesis of transfer functions by asymptotic fits in a warped frequency domain; (3) an investigation of the various substitution formulas, including the effects of nonlinearities; (4) use of pade approximation to the solution of the matrix exponential arising from the discrete state equations; and (5) an analytical integration of the state equation using interpolated input.

Parrish, E. A.↗
NASA NTRS1980

An equation of state for liquids

Closed expression for volume as function of pressure and temperature has been verified for over 250 liquids. Equation can assist chemical engineers, soild-state researchers, and others with interest in thermodynamic behavior of liquids.

Fedors, R. F.↗
NASA NTRS1983

Design and implementation of a multigrid code for the Euler equations

The steady-state equations of inviscid fluid flow, the Euler equations, are a nonlinear nonelliptic system of equations admitting solutions with discontinuities (for example, shocks). The efficient numerical solution of these equations poses a strenuous challenge to multigrid methods. A multigrid code has been developed for the numerical solution of the Euler equations. In this paper some of the factors that had to be taken into account in the design and development of the code are reviewed. These factors include the importance of choosing an appropriate difference scheme, the usefulness of local mode analysis as a design tool, and the crucial question of how to treat the nonlinearity. Sample calculations of transonic flow about airfoils will be presented. No claim is made that the particular algorithm presented is optimal.

Jespersen, D. C.↗
NASA NTRS2004

Sensitivity Equation Derivation for Transient Heat Transfer Problems

The focus of the paper is on the derivation of sensitivity equations for transient heat transfer problems modeled by different discretization processes. Two examples will be used in this study to facilitate the discussion. The first example is a coupled, transient heat transfer problem that simulates the press molding process in fabrication of composite laminates. These state equations are discretized into standard h-version finite elements and solved by a multiple step, predictor-corrector scheme. The sensitivity analysis results based upon the direct and adjoint variable approaches will be presented. The second example is a nonlinear transient heat transfer problem solved by a p-version time-discontinuous Galerkin's Method. The resulting matrix equation of the state equation is simply in the form of Ax = b, representing a single step, time marching scheme. A direct differentiation approach will be used to compute the thermal sensitivities of a sample 2D problem.

Hou, Gene↗
NASA NTRS1983

Robot arm dynamic model reduction for control

General methods are described by which the mathematical complexities of explicit and exact state equations of robot arms can be reduced to a simplified and compact state equation representation without introducing significant errors into the robot arm dynamic model. The model reduction methods are based on homogeneous coordinates and on the Langrangian algorithm for robot arm dynamics, and utilize matrix, vector and numeric analysis techniques. The derivation of differential vector representation of centripetal and Coriolis forces which has not yet been established in the literature is presented.

Bejczy, A. K.↗
NASA NTRS2004

Experiences on p-Version Time-Discontinuous Galerkin's Method for Nonlinear Heat Transfer Analysis and Sensitivity Analysis

The focus of this research is on the development of analysis and sensitivity analysis equations for nonlinear, transient heat transfer problems modeled by p-version, time discontinuous finite element approximation. The resulting matrix equation of the state equation is simply in the form ofA(x)x = c, representing a single step, time marching scheme. The Newton-Raphson's method is used to solve the nonlinear equation. Examples are first provided to demonstrate the accuracy characteristics of the resultant finite element approximation. A direct differentiation approach is then used to compute the thermal sensitivities of a nonlinear heat transfer problem. The report shows that only minimal coding effort is required to enhance the analysis code with the sensitivity analysis capability.

Hou, Gene↗
NASA NTRS1984

Discrete approximation methods for parameter identification in delay systems

Approximation schemes for parameter identification problems in which the governing state equation is a linear functional differential equation of retarded type are constructed. The basis of the schemes is the replacement of the parameter identification problem having an infinite dimensional state equation by a sequence of approximating parameter identification problems in which the states are given by finite dimensional discrete difference equations. The difference equations are constructed using linear semigroup theory and rational function approximations to the exponential. Sufficient conditions are given for the convergence of solutions to the approximating problems, which can be obtained using conventional methods, to solutions to the original parameter identification problem. Finite difference and spline based schemes using Paderational function approximations to the exponential are constructed, and shown to satisfy the sufficient conditions for convergence. A discussion and analysis of numerical results obtained through the application of the schemes to several examples is included.

Rosen, I. G.↗
NASA NTRS1973

Aircraft Range Optimization Using Singular Perturbations

An approximate analytic solution is developed for the problem of maximizing the range of an aircraft for a fixed end state. The problem is formulated as a singular perturbation and solved by matched inner and outer asymptotic expansions and the minimum principle of Pontryagin. Cruise in the stratosphere, and on transition to and from cruise at constant Mach number are discussed. The state vector includes altitude, flight path angle, and mass. Specific fuel consumption becomes a linear function of power approximating that of the cruise values. Cruise represents the outer solution; altitude and flight path angle are constants, and only mass changes. Transitions between cruise and the specified initial and final conditions correspond to the inner solutions. The mass is constant and altitude and velocity vary. A solution is developed which is valid for cruise but which is not for the initial and final conditions. Transforming of the independent variable near the initial and final conditions result in solutions which are valid for the two inner solutions but not for cruise. The inner solutions can not be obtained without simplifying the state equations. The singular perturbation approach overcomes this difficulty. A quadratic approximation of the state equations is made. The resulting problem is solved analytically, and the two inner solutions are matched to the outer solution.

Oconnor, Joseph Taffe↗
NASA NTRS1979

Multilevel modeling of the space construction base

A mathematical model representing a flexible space vehicle in terms of interconnected discrete masses is recast into hierarchical form to facilitate the application of multilevel control techniques. The model's state equations are temporarily decoupled by utilization of a set of coordination equations. A hierarchical form of the model is constructed with the set of coordination equations at the top of the hierarchy and the set of decoupled scalar state equations in the bottom level.

Chichester, F. D.↗
NASA NTRS1982

Integrating matrix solution of the hybrid state vector equations for beam vibration

A simple, versatile, and efficient computational technique has been developed for dynamic analysis of linear elastic beam and rod type of structures. Moreover, the method provides a rather general solution approach for two-point boundary value problems that are described by a single independent spatial variable. For structural problems, the method is implemented by a mixed state vector formulation of the differential equations, combined with an integrating matrix solution procedure. Highly accurate solutions are easily achieved with this approach. Example solutions are given for beam vibration problems including discontinuous stiffness and mass parameters, elastic restraint boundary conditions, concentrated inertia loading, and rigid body modes

Lehman, L. L.↗
NASA NTRS1974

Scaled parametric equation of state for oxygen in the critical region

Thermodynamic properties show an anomalous nonanalytic behavior in the critical region that cannot be represented by analytic equations used in engineering correlations. The formulation of scaling laws is discussed, taking into account an approach proposed by Schofield et al. (1969). A procedure for data analysis is described. The model considered has been found to give satisfactory results in tests conducted with a number of gases. The application of the described approach to density profiles is discussed. Attention is given to the model parameters, the critical exponents, and the coefficients for oxygen.

Levelt Sengers, J. M. H.↗
NASA NTRS1995

Determination of stability and control derivatives from the NASA F/A-18 HARV from flight data using the maximum likelihood method

This report is a compilation of PID (Proportional Integral Derivative) results for both longitudinal and lateral directional analysis that was completed during Fall 1994. It had earlier established that the maneuvers available for PID containing independent control surface inputs from OBES were not well suited for extracting the cross-coupling static (i.e., C(sub N beta)) or dynamic (i.e., C(sub Npf)) derivatives. This was due to the fact that these maneuvers were designed with the goal of minimizing any lateral directional motion during longitudinal maneuvers and vice-versa. This allows for greater simplification in the aerodynamic model as far as coupling between longitudinal and lateral directions is concerned. As a result, efforts were made to reanalyze this data and extract static and dynamic derivatives for the F/A-18 HARV (High Angle of Attack Research Vehicle) without the inclusion of the cross-coupling terms such that more accurate estimates of classical model terms could be acquired. Four longitudinal flights containing static PID maneuvers were examined. The classical state equations already available in pEst for alphadot, qdot and thetadot were used. Three lateral directional flights of PID static maneuvers were also examined. The classical state equations already available in pEst for betadot, p dot, rdot and phi dot were used. Enclosed with this document are the full set of longitudinal and lateral directional parameter estimate plots showing coefficient estimates along with Cramer-Rao bounds. In addition, a representative time history match for each type of meneuver tested at each angle of attack is also enclosed.

Napolitano, Marcello R.↗
NASA NTRS1973

Study of helium emissions from active solar regions

A theoretical study is made of the visible and UV line radiation of He I atoms and He II ions from a plane-parallel model flare layer. Codes were developed for the solution of the statistically steady state equation for a 30 level He I - II - III model, and the line and continuum transport equations. These codes are described and documented in the report along with sample solutions. Optical depths and some line intensities are presented for a 1000 km thick layer. Solutions of the steady state equations are presented for electron temperatures 10,000 to 50,000 K and electron densities 10 to the 10th power to 10 to the 14th power/cu cm.

Kulander, J. L.↗